A novel mathematical model of bone remodelling cycles for trabecular bone at the cellular level

Biomech Model Mechanobiol (2012) 11:973–982DOI 10.1007/s10237-011-0366-3

ORIGINAL PAPER

A novel mathematical model of bone remodelling cyclesfor trabecular bone at the cellular level

Bingbing Ji · Paul G. Genever · Ronald J. Patton ·Devi Putra · Michael J. Fagan

Received: 24 February 2011 / Accepted: 8 December 2011 / Published online: 5 January 2012© Springer-Verlag 2012

Abstract After an initial phase of growth and development,bone undergoes a continuous cycle of repair, renewal andoptimisation by a process called remodelling. This paperdescribes a novel mathematical model of the trabecular boneremodelling cycle. It is essentially formulated to simulatea remodelling event at a fixed position in the bone, inte-grating bone removal by osteoclasts and formation by oste-oblasts. The model is developed to construct the variationin bone thickness at a particular point during the remodel-ling event, derived from standard bone histomorphometricanalyses. The novelties of the approach are the adoption ofa predator–prey model to describe the dynamic interactionbetween osteoclasts and osteoblasts, using a genetic algo-rithm–based solution; quantitative reconstruction of the boneremodelling cycle; and the introduction of a feedback mecha-nism in the bone formation activity to co-regulate bone thick-ness. The application of the model is first demonstrated byusing experimental data recorded for normal (healthy) boneremodelling to predict the temporal variation in the number ofosteoblasts and osteoclasts. The simulated histomorphomet-ric data and remodelling cycle characteristics compare well

B. Ji · R. J. Patton · D. Putra · M. J. Fagan (B)Department of Engineering, University of Hull,Hull HU6 7RX, UKe-mail: [email protected]

B. Jie-mail: [email protected]

R. J. Pattone-mail: [email protected]

D. Putrae-mail: [email protected]

P. G. GeneverDepartment of Biology, University of York, York, UKe-mail: [email protected]

with the specified input data. Sensitivity studies then revealhow variations in the model’s parameters affect its output;it is hoped that these parameters can be linked to specificbiochemical factors in the future. Two sample pathologicalconditions, hypothyroidism and primary hyperparathyroid-ism, are examined to demonstrate how the model could beapplied more broadly, and, for the first time, the osteoblastand osteoclast populations are predicted for these conditions.Further data are required to fully validate the model’s predic-tive capacity, but this work shows it has potential, especiallyin the modelling of pathological conditions and the optimi-sation of the treatment of those conditions.

Keywords Mathematical model · Trabecular bone ·Osteoclast · Osteoblast · Remodelling

1 Introduction

Bone is a remarkable dynamic tissue, which continuouslyrepairs, renews and adapts itself to its environment and tomaintain its function as the body’s structural support and min-eral reservoir (Parfitt 1994; Sommerfeldt and Rubin 2001).These dynamic behaviours are achieved through the remod-elling process, which occurs at the basic multicellular unit(BMU) (Frost 1986). Bone remodelling is a coupled pro-cess of bone resorption, carried out by osteoclasts, and boneformation, carried out by osteoblasts. The balance betweenthe volume of resorbed bone and of the newly formed boneand the activation frequency of BMUs determines the integ-rity of the bone structure and its strength throughout its life(Christiansen 2001; Seeman and Delmas 2006).

After initial bone formation, which begins in utero andpersists throughout adolescence until skeletal maturity, bone

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continues to renew itself and adapt its structure and materialproperties to the external mechanical demands and biologi-cal environment via a localised process called ‘bone remod-elling’ (Einhorn 1996). In a normal adult’s bone, the balancefrom remodelling is approximately zero and the mean acti-vation frequency about 0.33 per year for trabecular bone(Eriksen et al. 1985, 1986b), where Eriksen defines activa-tion frequency as the formation rate of a new remodellingcycle at a particular point. However, the remodelling bal-ance becomes negative through the ageing process and dis-use (Zaidi 2007). Bone loss appears to begin between ages18 and 30 years, but the process is slow because the acti-vation frequency is so low (Gilsanz et al. 1988). However,pathologies, such as hyperthyroidism, oestrogen deficiency,thyrotoxicosis and hypogonadism, can speed up the bone-loss process (high-turnover osteoporosis) causing structuraldamage, decreased bone strength and increased fracture risk(Seeman and Delmas 2006; Eriksen et al. 1985; Zaidi 2007).There are also some pathological conditions such as hypo-thyroidism, which can induce a positive remodelling balance(Eriksen et al. 1986a). In this case, the positive balance isdue to the significant reduction of bone resorption comparedto the normal condition, while the bone formation remainssimilar.

The BMU cycle suggests that osteoporosis and other bone-loss diseases can be treated by the repeated use of selectedagents that affect different parts of the remodelling processand thereby create an overall incremental gain in bone mass(Frost 1979). For example, the use of therapeutic agents toincrease BMU activations in conjunction with a reduction inosteoclast activity while maintaining osteoblast activity willlead to an accumulation of new bone, in a similar way to theprocess of hypothyroidism. This has led to the developmentof antiresorptive therapies for post-menopausal bone loss,including oestrogens, the selective oestrogen receptor mod-ulator (SERM) tamoxifen and raloxifene, bisphosphonatesand calcitonin (Zaidi 2007; Rodan and Martin 2000). Thesedevelopments suggest that a comprehensive understandingof the remodelling process at the BMU level could lead tobetter methods for manipulating the remodelling cycle forthe treatment of bone-loss diseases.

Recent reviews of the bone remodelling process haverevealed a growing number of factors that are involved inits regulation (Zaidi 2007; Allori et al. 2008). These includeautocrine and paracrine signalling molecules, systemic hor-mones and extracellular matrix components that affect cell-to-cell communication, migration, adhesion, proliferationand differentiation. Most of those findings are obtained fromisolated observations of either in vitro studies or in vivoexperiments using genetically manipulated animals. Thesefindings have shown that osteoblasts are able to regulate theactivity of osteoclasts, for example, expression of receptoractivator of NF κB ligand (RANKL) by osteoblasts directly

interacts with RANK on osteoclast progenitors to drive osteo-clastogenesis. This process also depends on the level of osteo-protegerin (OPG) that can act as a soluble decoy receptor forRANKL, thereby inhibiting RANK-mediated osteoclasto-genesis (for a review, see Boyce and Xing 2008). However,bone remodelling is a very complex, integrated process, andthese localised findings may give limited information aboutthe overall effects of those factors on the bone remodellingprocess. This highlights the need for tools, which can inte-grate these partial observations to sets of rules that define thebehaviour of this complex system. Such rules would enableone to synthesise agents that are able to manipulate the bal-ance and activation frequency of bone remodelling cycles.

Mathematical modelling has proven to be a powerful toolfor knowledge synthesis and analysis of complex biologi-cal systems. However, there have been few attempts to inte-grate the growing knowledge of the factors affecting thebone remodelling process into mathematical models that areable to mimic the integrated process of bone remodellingcycles at the BMU level. A mathematical model describ-ing the dynamic interaction between pre-osteoblasts, oste-oblasts and osteoclasts in response to parathyroid hormone(PTH) stimulation has been reported (Kroll 2000), whichwas subsequently extended to include the effect of oestro-gen stimulation on the dynamics of osteoblast and osteoclastpopulations (Rattanakul et al. 2003). The first mathematicalmodel to mimic the temporal dynamics of bone remodellingcycles at a single BMU was proposed by Komarova et al.(2003). In addition to the dynamic interaction between osteo-clasts and osteoblasts, influenced by the effects of autocrineand paracrine regulations at a BMU, this also included thedynamics of the bone mass during the resorption and for-mation processes. Moroz et al. (2006) extended this modelby including the osteocyte population to accommodate theeffects of mechanical loading. A dynamic interaction modelof pre(responding)-osteoblasts, osteoblasts and osteoclastsincluding the RANK-RANKL-OPG pathway has also beenreported (Lemaire et al. 2004). This was developed furtherto investigate the effects of different mechanisms in mod-elling the RANK-RANKL-OPG pathway by Pivonka et al.(2008), and more recently by the same group to investigatethe pathway-mediated bone diseases and treatment strate-gies for imbalance in the pathway (Pivonka et al. 2010). Inorder to obtain a better understanding of the bone remodel-ling process, Ryser et al. (2010) developed a spatiotempo-ral model to simulate the dynamics of bone cell populationsas well as RANKL and OPG for a trabecular BMU (Ryseret al. 2010). Following this, another spatiotemporal contin-uum model integrating some of the most important existingpathways has also been proposed to reproduce the spatio-temporal dynamics of individual cortical BMUs (Buenzliet al. 2011). It is important to note that among the modelsmentioned above, only the models developed by Komarova,

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Moroz, Ryser and Buenzli mimicked the temporal dynamicsof the bone remodelling process at a single BMU.

In this paper, we propose a novel mathematical modeldescribing the temporal dynamic interaction between osteo-clasts and osteoblasts at a single BMU with their correspond-ing bone resorption and formation activities. The model isdeveloped to mimic the variation in bone thickness at a par-ticular point during the bone remodelling process in bothnormal and pathological conditions, which are obtained fromhistomorphometric analysis (e.g., Agerbaek et al. 1991; Erik-sen et al. 1984b, 1985, 1986a) and to replicate the observeddynamic interaction between osteoclasts and osteoblasts dur-ing the remodelling process. We refer to these as recon-structed remodelling cycles (or curves).

There are three novelties of the proposed model. Firstly,the adoption of a predator–prey model to replicate the sequen-tial dynamic interaction between osteoclasts and osteoblastsat a BMU; secondly, the bone remodelling cycles are recon-structed quantitatively for the first time; and thirdly, a feed-back mechanism is used to maintain the balance of bonethickness during a remodelling cycle. The feedback mech-anism allows decreasing bone formation rates, as observedby histomorphometric analysis, but neglected in the model ofKomarova et al. (2003), and removes the artificial assumptionmade by them about the steady-state osteoblast and osteoclastpopulations at a single BMU in order to obtain the quiescentperiod.

2 Model

The model considered in this paper simulates the dynamicinteraction between osteoclasts and osteoblasts and their cor-responding resorption and formation activities at a BMU dur-ing the bone remodelling process, where the interaction isbased on the competition model of a predator–prey system(Gause et al. 1936; Kuang 1990). The motivation to adoptthe predator–prey model is based on its key characteristic ofcompetitive cyclic growth between the prey and the predatorpopulation, and the fact that the population cannot decreaseto negative values. These properties are similar to the growthof osteoclasts, which is tightly coupled to the growth of oste-oblasts during the remodelling process at a BMU (Parfitt2000; Udagawa et al. 2006). Differentiation into the osteo-clastic and osteoblastic lineages involves several intermedi-ate stages (e.g. the osteoclast lineage develops from hemato-poietic precursor cells through monocyte differentiation andfusion to osteoclast formation (Roodman 1999; Teitelbaum2000), while the osteoblast lineage arises from mesenchymalstem cells through to pre-osteoblasts, mature bone-formingosteoblasts, osteocytes and bone lining cells (Aubin 1998a)).In the current model, the terms ‘osteoclast’ and ‘osteoblast’include both precursor and mature cells. Thus, the rate of

change in cell populations includes the production of precur-sors, the formation of mature cells and their eventual removalor transformation.

Based on this definition, the model proposes that the osteo-clast–osteoblast interaction is given by the following set ofdifferential equations:

xoc(t) =(

a − b√

xob(t))

xoc(t) (1)

xob(t) =(

cx 2

oc(t)

Koc + x 2oc(t)

− d

)xob(t) (2)

where xoc(t) and xob(t) are the osteoclast and osteoblastpopulations, respectively; xoc(t) = dxoc(t)/dt is the var-iation of xoc(t) with time, for example; and a, b, c, d andKoc are unknown scalar parameters. The model defined byEqs. (1) and (2) belongs to the class of Gause-type preda-tor–prey models (Gause et al. 1936). Its selection was notbased on any specific underlying biological mechanisms,except the requirement to replicate the dynamics between theosteoblasts and osteoclasts. However, following a detailedinvestigation of the results and a parameter sensitivity study,relationships between the parameters and some biologicalfactors do become evident, as discussed later. It can be shownthat all solutions for xoc(t) and xob(t) result in periodic orbits(Kuang 1990) and further demonstrated below for all pos-itive initial conditions (i.e., xoc(0), xob(0) > 0), except atits unique positive equilibrium point. This property guaran-tees the periodicity of remodelling cycles and the couplingbetween osteoclast and osteoblast population growth. How-ever, note that the factors activating the bone remodellingcycle, such as biological and mechanobiological signals, arenot included in the model, and the periodicity of the modeldoes not correspond to one single bone remodelling cycle, butrather reflects the average of many bone remodelling cycles.

The model for the bone resorption and formation activitiesis proposed as:

D(t) = Fform(t) × Ffeedback(t) − Fres(t) (3)

where D(t) represents the instantaneous cavity depth cre-ated by a BMU during one single bone remodelling cycle,Fform(t) and Fres(t) are the bone formation and resorptionrates, and Ffeedback(t) represents a feedback mechanism toco-regulate the bone formation during bone remodelling. Theinitial value of D(t) is zero. D(t) then becomes negative dur-ing the resorption phase, before returning to zero or finishingwith a negative or positive value, depending on whether thereis net bone resorption or formation. D(t) = d D(t)/dt rep-resents the variation of that cavity depth with time.

The bone resorption and formation rates are dependent onthe resorptive and formative activities of the osteoclasts andosteoblasts and population of cells (Lemaire et al. 2004), andit is assumed that each cell type has the same and constantlevel of activity during one bone remodelling cycle (Rodan

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1998; Lemaire et al. 2004). Based on these assumptions, thebone resorption and formation rates are then only related tothe population of osteoclasts and osteoblasts. The functionsto define these relationships and produce qualitatively rea-sonable resorption depth evolution curves are proposed asfollows:

Fform(t) = exob(t)

Kob + xob(t)(4)

Fres(t) = f x2oc(t) (5)

where e, f and Kob are also unknown scalar parameters. Theequation describing the feedback mechanism is defined as:

Ffeedback(t) = exp((D − D(t))/DM ) (6)

D is defined as the reference value of D(t) and equals the bal-ance between the cavity depth resorbed by osteoclasts and thedepth refilled by osteoblasts during one BMU remodellingcycle (D = maximum formation height−maximum resorp-tion depth). The term (D − D(t)) represents the cavity depthneeded to be refilled and is normalised by its maximal valueDM (DM actually equals the maximum cavity depth resorbedby osteoclasts during one BMU remodelling cycle). Thefeedback mechanism is designed to sense the remaining cav-ity depth refilled by osteoblasts during one BMU remodel-ling cycle and then regulate the bone formation rate. Theproposed equations of feedback mechanism can satisfy thisrequirement completely: that is, at the beginning of boneformation period, the term (D − D(t)) reaches its maximalvalue, since the cavity has not been refilled at all, and the feed-back mechanism outputs its maximum; as the bone formationproceeds, the value of the term (D − D(t)) decreases as wellas Ffeedback(t), since more of the cavity is being refilled.

The feedback mechanism ensures that the rate of bonematrix formation is related to the current cavity depth. Thisallows the model to exhibit the observed phenomenon that theapposition rates are large at the start of the formation periodand decrease gradually towards zero at the end (Eriksen et al.1984b, 1985, 1986b).

Using Eqs. (4) to (6), Eq. (3) can then be reformulated as:

D(t) = exob(t)

Kob + xob(t)exp((D − D(t))/DM ) − f x 2

oc(t)

(7)

Thus, the model for bone remodelling is based on Eqs. (1),(2) and (7), which are solved using the Matlab computationalsoftware package (v7.7.0, Mathworks, Natick, USA; withthe Runge–Kutta integration method ode45 and a specifiedtolerance of 10−10).

In our model, there are eight parameters in the equationsincluding a to f, Kob and Koc, which directly affect the solu-tions to these equations. Different combinations of param-eters correspond to various biochemical conditions wherebone remodelling happens (i.e. as shown later in Table 5 for

Table 1 Histomorphometry data for normal trabecular bone remodel-ling, as reported in the literature and output from the new model

Phenomena Experimental Source Model(mean) values outputs

Maximum resorption depth 62µm [1] 62.2µm

Resorption period 48 days [1] 50.8 days

Maximum resorption rate 3.9µm/day [1] 3.8µm/day

Maximum formation height 62µm [2] 62.2µm

Formation period 145 days [2] 145 days

Maximum formation rate 2.1µm/day [2] 1.4µm/day

Quiescent period 902 days [3] 901.4 days

Maximum osteoclastpopulation

8.0 cells (est) [4,5] 9.0 cells

Maximum osteoblastpopulation

2,000 cells (est) [4,6] 1,930.7 cells

Only the first seven quantities of experimental data are used to calcu-late the model parameters, and the cell populations are estimated fromvalues reported for Haversian remodelling. [1] Eriksen et al. (1984a);[2] Eriksen et al. (1984b); [3] Eriksen et al. 1986; [4] Parfitt (1994); [5]Jaworski et al. (1981); [6] Jaworski and Hooper (1980)

the normal condition, hypothyroidism and hyperparathyroid-ism). In order to find the parameters corresponding to thesevarious conditions, a genetic algorithm approach is used tosearch for the parameters’ values in the parameter space.

Several steps are required to calculate the parameters’ val-ues for each condition. Firstly, nine phenomena are normallyused to characterise the normal bone remodelling cycle, seeTable 1, which are defined as follows:

Resorption period = tres − tinitial

Maximum resorption depth = D(tinitial) − D(tres)

Formation period = tform − tres

Maximum formation height = D(tform) − D(tres)

Quiescent period = tend − tform

Maximum resorption rate = Max(Fres(t)), t

∈ [tinitial, tres]

Maximum formation rate = Max(Fform(t)

×Ffeedback(t)), t ∈ �tres, tform�Maximum osteoclast population = Max(xoc(t))

Maximum osteoblast population = Max(xob(t))

where, as shown in Fig. 1, tinitial and tres represent thetimes when the bone resorption phase begins and ends,respectively; tform and tend are the times when the boneformation phase and quiescent phase ends, respectively;D(tinitial), D(tres) and D(tform) are the cavity depth corre-sponding to times tinitial, tres and tform; Fres(t) and Fform(t)×Ffeedback(t) represent the rates of bone resorption and forma-tion (some of these variables are marked in Fig. 1).

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A novel mathematical model 977

Fig. 1 Model simulations of the variation in osteoclast and osteo-blast populations and bone thickness during the normal bone remod-elling cycle (note the osteoblast scaling factor). (1: (tinitial, Dinitial), 2:(tres, Dres), 3: (tform, Dform), 4: (tend, Dend))

The published experimental data are incomplete for pri-mary hyperparathyroidism and hypothyroidism; in particu-lar, the osteoclast and osteoblast populations are not reported(see Table 4). Therefore, only the first seven quantities inTable 1 (and later Table 4) are used to characterise the nor-mal, hypothyroidism and hyperparathyroidism conditions inour model. However, this has the important advantage thatthe osteoclast and osteoblast populations predicted by themodel for the normal condition can be used to confirm thevalidity of the approach.

The genetic algorithm searches the solution space to mini-mise the difference between the model outputs and publishedexperimental data. Thus, the function that is minimised is:

F(X) =∑

i=1:7abs((P(X)i − Ei )) (8)

X = [a, . . . , f, Kob, Koc] (9)

where X = [a, . . . , f, Kob, Koc] is a row vector consist-ing of the eight parameters in the model equations and rep-resents one point in the parameter space; and P(X)i andEi (i = 1, . . .7) represent the model outputs correspond-ing to each point in the parameter space and the exper-imental values of the first seven phenomena in Tables 1and 4, respectively. (For more specific details on thisaspect, see http://www.mathworks.com/help/toolbox/gads/f6010dfi3.html.) Once the genetic algorithm has identified

the optimum parameter values, Eqs. (1), (2) and (7) are usedto calculate the detailed variation of xoc(t), xob(t) and D(t)with time.

In the following application of the model, the normal boneremodelling cycle is reconstructed, and the sensitivity of theresults to the model parameters investigated. Then, two path-ological conditions, primary hyperparathyroidism and hypo-thyroidism, are examined to demonstrate how the model canbe used to simulate other abnormal conditions.

3 Results

3.1 The normal bone remodelling cycle

Table 1 presents published experimental data for the normaltrabecular bone remodelling cycle, from which the parametervalues shown in Table 2 have been calculated using a geneticalgorithm approach. These parameters were used to recalcu-late the histomorphometric values also included in Table 1,which, with the exception of the maximum formation rate,match the experimental data closely. While we would expectthe first seven quantities to match closely, remember thatthe maximum osteoclast and osteoblast populations were notused in the solution process, and yet the values predicted bythe model agree well with the experimental data. However,it should be noted that the cell populations quoted in Table 1are estimated from experimental measurements for Haver-sian remodelling, where we assume that the trench-shapedremodelling volume in cancellous bone is approximately halfof that of the tunnel-shaped remodelling volume of corticalbone (Jaworski and Hooper 1980; Jaworski et al. 1981; Parfitt1994).

In the simulation, the initial conditions were set toxoc(0)= xob(0)= 0.1 so that the osteoclast and osteoblastpopulations were approximately zero at the start of theresorption period. Sensitivity studies (not included here)revealed that the simulations were only marginally affected

Table 2 Parameter values derived by the model for the normal remod-elling cycle

Parameter Value

a 0.0558 day−1

b 0.0065 cells−1/2 day−1

c 1.82 × 109 day−1

d 0.0099 day−1

e 17.2 µm−1

f 0.0461 cells−2µm−1

Koc 2.44 × 1011 cells2

Kob 4.32 × 104 cells

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Fig. 2 The periodic orbit of the osteoclast–osteoblast interaction dur-ing the normal bone remodelling cycle, which proceeds in an anticlock-wise direction

by the initial choice of these parameters. According to BMUtheory and experimental observations, there should not beany osteoclasts and osteoblasts active during the quiescentperiod (Parfitt 1994). Therefore, due to the requirement in thesimulations to define xoc(t) and xob(t) in the model (Eqs. (1)and (2)) as real (rather than integer) terms, any values below0.5 were regarded as zero (i.e., no active osteoclast or osteo-blast). The end of the resorption period (and start of the for-mation period) was defined when the number of osteoclastsfell below 0.5. However, because the maximum number ofosteoblasts is typically two orders of magnitude greater thanthe maximum number of osteoclasts, the end of the forma-tion period (and the start of the quiescent period) was definedas the moment when the formed cavity depth D(t) reached99.5% of its maximal value (as shown in Fig. 1), rather thanthe time when the osteoblast population fell below 0.5. Exam-ination of the osteoblast population (Fig. 1) shows that, in themodel, there are still a significant number of non-active oste-oblasts present when 99.5% of the cavity is refilled, whichtake significantly longer to decay. The reason for this incon-sistency lies in the underlying predator–prey equations, inwhich the preys (osteoclasts) thrive again as soon as thepredators (osteoblasts) have decreased, and vice versa, notafter a period where neither population is present. Main-tenance of so many inactive osteoblasts is physiologicallyunlikely, which highlights a limitation of the predator–preyapproach.

The cyclic variation in osteoblast and osteoclast popula-tions and bone thickness simulated by the model are pre-sented in Fig. 1, and the dynamic interaction between theosteoclasts and osteoblasts are shown in detail in Fig. 2. Thelatter shows how the remodelling cycle commences withthe growth of the osteoclast population in the absence ofosteoblasts. As the osteoclast population continues to grow,

the osteoblast population also starts to increase; then, as theosteoblast growth increases further, the osteoclast numbersstart to decline. While the osteoclast population is decreasing,the osteoblast population grows faster but starts to decreaseagain as the osteoclasts completely disappear. Note that Fig. 2does not reflect the difference in the rates at which these pop-ulations change and in particular the much shorter period ofosteoclast activity compared to the longer period of osteo-blast activity. However, this can be seen in Fig. 1, whichshows, for example, the rapid decline of the osteoclast pop-ulation as the number of osteoblasts increases.

3.2 Sensitivity to the model parameters

The sensitivity of the simulations to variations in parametersa, b and Koc in Eqs. (1) and (2) is shown in Fig. 3. The devi-ation in the maximum number of osteoclasts and osteoblastsis presented when the parameters are varied between 0.5 and1.5 of the ‘normal’ value. This sensitivity analysis shows thatparameter a affects both the growth of osteoclasts and osteo-blasts; parameter b affects mainly the growth of osteoblasts;while parameter Koc affects only the growth of osteoclasts.Parameter d (not shown) influences the apoptosis or elimi-nation rate of osteoblasts (Eq. 2), and an increase in its valuewill speed up their removal. The parameter Kob in Eq. (7) isresponsible for controlling the maximum formation rate. Thevalues of parameters c, e and f are assumed to be constantin all simulations, because c and e are dependent on Koc andKob, respectively, and f is a scaling coefficient for the bone

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

1

2

3

4

5

6

Normalized parameter variation ratio

Rat

io o

f ce

ll nu

mbe

rs

a:OCa:OBb:OCb:OBKoc:OCKoc:OB

Fig. 3 The effects of variation in the parameters on the osteoblastand osteoclast populations, normalised to the values of the base case(as defined in Tables 1 and 2), where each parameter was varied individ-ually. For example, the lines a:OB and a:OC demonstrate the variation inosteoblast and osteoclast populations as parameter ‘a’ is varied between0.5 and 1.5 times its base value

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Table 3 Summary of the relationship between the model parametersand remodelling activity

Parameters Observations

a ↑ Osteoclasts ↑ osteoblasts ↑b ↓ Osteoclasts – osteoblasts ↑↑d ↑ Osteoclasts – osteoblasts↓Koc ↑ Osteoclasts ↑ osteoblasts –

Kob ↑ Formation rate ↓↓ = decrease,↑ = increase,↑↑ = significant increase,− = little orno effect

resorption. A summary of the sensitivity results is includedin Table 3.

3.3 Simulation of the remodelling cycles in pathologicalconditions

Once the basic equations describing the remodelling cycleare established, they can be modified to simulate pathologicalconditions, and with that information, the opportunity thenarises to examine the effect of different therapies. For exam-ple, histomorphometric analyses of trabecular bone samplesfrom 19 primary hyperparathyroid (PHPT) patients (6 menand 13 women) and 18 hypothyroid (HT) female patientshave been reported (Eriksen et al. 1986a,b); the key data areincluded in Table 4.

The controlling parameters for these two pathologies werecalculated using the same procedures as outlined above andare compared to those of the normal remodelling cycle inTable 5. The resulting remodelling cycles for the two condi-tions are shown in Figs. 4 and 5, for the PHPT and HT condi-tions, respectively. The basic shapes of the curves are similarto those for normal remodelling (Fig. 1) in both cases, butthe maximum number and ratio of osteoclasts and osteoblasts

Table 5 Comparison of parameter values derived by the model fornormal and disease conditions

Parameters Hypothyroidism Normal P. hyperpara-thyroidism

a 0.0327 day−1 0.0558 day−1 0.1231 day−1

b 0.0044 cells−1/2 0.0065 cells−1/2 0.0129 cells−1/2

day−1 day−1 day−1

ca 1.82 × 109 day−1

d 0.0025 day−1 0.0099 day−1 0.0150 day−1

ea 17.2µm−1

f a 0.0461 cells−2

µm−1

Koc 1.81 × 1011 cells2 2.44 × 1011 cells2 1.74 × 1011 cells2

Kob 9.64 × 104 cells 4.32 × 104 cells 6.42 × 104 cells

a Parameters c, e and f are assumed to be constant

are different in each case, as is the period of the remodellingcycle. The actual histomorphometric data calculated by theseparameter values are compared to the original clinical val-ues in Table 4. The experimental and model data generallycompare well, although there are differences in the resorp-tion and formation rates and an 18% difference in the PHPTresorption period, but the resorption depth is still correct.However, the model is also able to predict the osteoclast andosteoblast populations for each condition, which were notpreviously reported in the literature. The predicted osteo-blast and osteoclast cell populations in the PHPT conditionare nearly double those of the HT conditions, with the num-ber of PHPT osteoclasts and osteoblasts 26 and 29%, respec-tively, higher than the ‘normal’ case (Table 1). As a result, theosteoblast/osteoclast ratio for these two conditions is 219.8and 233.2, respectively, while the ratio for normal remodel-ling is 214.5.

Table 4 Comparison of the experimental and simulated remodelling cycles in primary hyperparathyroidism and hypothyroidism; with experimentaldata from Eriksen et al. (1986a,b)

Phenomena P. hyperparathyroidism Hypothyroidism

Experimental (mean) values Model outputs Experimental (mean) values Model outputs

Maximum resorption depth (µm) 45.2 45.2 42.1 42.1

Resorption period (days) 31 25.4 76 75.9

Maximum resorption rate (µm/day) 5.7 5.9 2.9 1.4

Maximum formation height (µm) 45.2 45.2 59.0 59.0

Formation period (days) 172 172 620 625.9

Maximum formation rate (µm/day) 2.2 1.1 0.74 0.3

Quiescent period (days) 390 393 2,098 2,068.6

Maximum osteoclast population (cells) n/a 11.3 n/a 5.5

Maximum osteoblast population (cells) n/a 2,483.8 n/a 1,280.1

n/a = not available. Only the first seven quantities of experimental data are used to calculate the model parameters

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0 100 200 300 400 500 6000

5

10

Time [day]

Num

ber

of C

ells osteoclast

osteoblast (*200)

0 100 200 300 400 500 600

-40

-20

0

Time [day]

Bon

e T

hick

ness

[µm

]

Fig. 4 Model simulations of the variation in osteoclast and osteoblastpopulations and bone thickness during the remodelling cycle with pri-mary hyperparathyroidism (note the osteoblast scaling factor)

0 500 1000 1500 2000 25000

2

4

6

Time [day]

Num

ber

of C

ells osteoclast

osteoblast(*200)

0 500 1000 1500 2000 2500

-40

-20

0

20

Time [day]

Bon

e T

hick

ness

[µm

]

Fig. 5 Model simulations of the variation in osteoclast and osteoblastpopulations and bone thickness during the bone remodelling cycle withhypothyroidism (note the osteoblast scaling factor)

4 Discussion

Bone remodelling is most conveniently considered at thebasic multicellular unit (BMU) level, which integrates theosteoclastic removal and osteoblastic formation processes.The sequence of activities that take place during remod-elling occur over an extended timeframe, for example,typically 200 days for normal remodelling (Eriksen et al.1984a,b) followed by a quiescent period of possibly 900 days

(Eriksen et al. 1986b). In the ideal situation, the volume ofbone removed and deposited will be the same, but whetherthis occurs in reality will depend on many complex factors. Inthe simplest terms, it relies on the number of cells involvedand the period and rate of the cellular resorption and for-mation processes. A change in any of these will lead to avariation in the remodelling outcome and a net loss or gainin bone volume.

The aim of the mathematical model is to simulate the activ-ity and interactions between the cells and resultant effect onthe bone. A predator–prey-based mathematical relationshipis used to define the basic associations, although the gapsbetween bouts of activity are much longer than would nor-mally be expected in predator–prey situations. In this firstapplication of the model, the model parameters are initiallyestablished for normal (healthy) remodelling, with the resul-tant governing equations then shown to provide a reasonablesimulation of the observed cellular activities, with the pre-dictions of peak osteoclast and osteoblast populations com-paring well to the estimated populations. However, it shouldbe noted that because these primary data are limited andcome from a number of different sources, the values of allthe model parameters needs to be treated with some cau-tion. Also, inevitably, there will be some statistical variationin the experimental input data, the effect of which needs fur-ther detailed investigation when more robust primary data areavailable. In future, this variability could be included auto-matically in the solution phase of the model to provide anenvelope of bone remodelling behaviour.

The potential for the model to be used in investigations ofthe bone remodelling cycle and the effects of different path-ological conditions is demonstrated by considering primaryhyperparathyroidism (PHPT) and hypothyroidism (HT). Thenatural histories of these two conditions are quite differentas demonstrated in Table 4 (Eriksen et al. 1986a,b), but themodel allows both the remodelling cycles to be reconstructedand the complex temporal interaction between the osteoblastsand osteoclasts and the resultant effect on bone thickness tobe demonstrated. Unlike the ‘normal’ case, the numbers ofcells involved in BMU remodelling in these two scenariosis not reported in the literature, but they can be predicted bythe simulation. Despite the fact that the two cases are verydifferent (e.g. the formation periods are 390 and 2,098 daysfor PHPT and HT, respectively) and the predicted numbersof cells involved in each are quite different, it is interesting tonote that the osteoblast/osteoclast ratios are similar. In fact,in all three cases, the ratio ranges from 215 to 233. Unfortu-nately, at the present time, there are no other data availablein the literature to confirm these predictions.

In reality, bone remodelling activity is initiated and regu-lated by molecular reactions and processes. In an attempt toexamine the sensitivity of the results to the model’s param-eters and thereby identify possible relationships with these

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A novel mathematical model 981

biochemical factors, the effects of parameter variations onthe remodelling process and results were examined. Theresults of these sensitivity studies are presented in Fig. 3,with suggestions of relationships between the parametersand, for example, cell populations shown in Table 3. Howthe underlying regulatory factors involved in the remodellingprocess relate to these individual model parameters needsmuch more investigation. But, for example, the RANKL-RANK-OPG pathway constructs a basic control networkof bone remodelling, and many signalling cues, such aslocally acting growth factors and systemic hormones, reg-ulate bone remodelling via the RANK-RANKL-OPG path-way (Canalis 1993; Manolagas 2000). Thus, the parame-ter a that positively controls the production of osteoclastsmay reflect RANKL levels, while parameter b that nega-tively regulates the production of osteoclasts may be relatedto OPG. (However, note in reality these parameters arelikely to linked, since RANKL and OPG do not have inde-pendent effects on osteoclasts, with OPG able to inhibitthe production of RANKL.) Since thyroid hormone is apotent stimulator of osteoclasts and osteoblasts (Eriksen et al.1986a), we would expect a significant decline in their num-bers in hypothyroidism (HT), and indeed this is seen to bethe case (Table 4). Consideration of the parameters calcu-lated for HT in Table 5 shows that both parameters a andb decrease compared to the normal case, from which wepredict that the levels of RANKL and OPG are similarlydecreased. This is confirmed, to some degree, by Kanataniet al. (2004) who observed a decline in OPG with this condi-tion, but no change in RANKL; however, Miura et al. (2002)indicated RANKL level can also be decreased under thiscondition

The predator–prey model was selected to replicate thegeneral dynamics between the osteoblasts and osteoclasts,rather than any specific underlying biological mechanisms.However, the application of the model shows that, based onthe limited experimental data available, it can produce rea-sonable simulations of the remodelling cycles in trabecularbone. Clearly, much more work is required to validate themodel, but the calculated histomorphometric data and remod-elling cycles compare well with the sample input data. Webelieve that these show that the model has merit and predic-tive potential, especially in the future modelling of patholog-ical conditions and the optimisation of the treatment of thoseconditions.

Acknowledgments We would like to thank an anonymous reviewerwho provided very detailed and constructive feedback on earlier ver-sions of the manuscript. This work was partly supported by the UKEngineering and Physical Sciences Research Council through grantEP/E057365/1 and the CSC (China Scholarship Council).

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